Curvature of a circle approaches zero as radius goes to infinity

In summary, the conversation discusses the mathematical expression for the relationship between the radius and curvature of a circle. It is concluded that as the radius tends towards infinity, the curvature of the circle tends towards zero. This can be expressed using the limit expression lim 1/r with r-> infinity. It is also mentioned that the value of pi remains constant even for circles with infinite radius. The concept of indeterminate forms is also brought up in relation to evaluating limits of functions.
  • #1
BOAS
552
19
Hello,

this isn't a homework problem, so I'm hoping it's okay to post here.

I would like to know the correct way to mathematically express the idea in my title. It is intuitively obvious that as the radius of a circle increases, it's curvature decreases.

I looked it up and found that the curvature of a circle is equal to the reciprocal of it's radius. Certain assumptions are often made when looking at lenses, i.e the wave fronts reaching the lens are parallel, or have 0 curvature - In other words, the object distance is infinitely far away.

But, 1/∞ ≠ 0

So how do I express it properly?

In words, I think it goes something like this - As the radius tends towards infinity, the curvature of the circle tends towards zero.
 
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  • #2
Wouldn't you just use the lim 1/r expressions with r-> infinity to express it?
 
  • #3
jedishrfu said:
Wouldn't you just use the lim 1/r expressions with r-> infinity to express it?

That would be my guess but I'm unsure of how to formulate that.

[itex]lim_{r \rightarrow ∞} \frac{1}{r} = 0[/itex]

Like that?
 
  • #5
If you imagine a circle with infinite radius, then its circumference is also infinite.
Then what would be the value of pi be? Infinite divided by infinite. Can you say what
it is?
I think the real projective line may be a picture of this kind of "circle":
http://en.wikipedia.org/wiki/Real_projective_line
 
  • #6
7777777 said:
If you imagine a circle with infinite radius, then its circumference is also infinite.
Then what would be the value of pi be?
The same as always. ##\pi## is a constant (its value never changes).
7777777 said:
Infinite divided by infinite. Can you say what
it is?
No. There are several indeterminate forms, including [∞/∞], [0/0], [∞ - ∞], and a few others. These are indeterminate, because you can't determine a value for them.

They usually come up when we are evaluating limits of functions.
7777777 said:
I think the real projective line may be a picture of this kind of "circle":
http://en.wikipedia.org/wiki/Real_projective_line
 

1. What is the meaning of "curvature of a circle approaches zero"?

The curvature of a circle is a measure of how much the curve deviates from being a straight line. When we say that the curvature approaches zero, it means that the circle becomes more and more like a straight line as its radius increases.

2. How is the curvature of a circle calculated?

The curvature of a circle is calculated by dividing the circle's circumference by its diameter. This value is equal to 1/r, where r is the radius of the circle.

3. Why does the curvature of a circle approach zero as the radius goes to infinity?

This is because as the radius of a circle increases, the circle becomes larger and larger, making the curvature of the circle decrease. In the limit of an infinite radius, the curvature becomes infinitesimally small, approaching zero.

4. What is the relationship between radius and curvature in a circle?

The relationship between radius and curvature in a circle is inverse. This means that as the radius of a circle increases, the curvature decreases, and vice versa.

5. How does the concept of "curvature of a circle approaches zero" relate to real-world applications?

This concept is important in various fields such as mathematics, physics, and engineering. For example, in calculus, it is used to understand limits and derivatives. In physics, it is used to study the motion of objects in circular motion. In engineering, it is used to design curved structures such as bridges and roads.

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